“I wish we didn’t have to ask,” she said. “But to me, that’s the rock and the hard place.”

Race is difficult to count because, unlike income or employment, it is a social category that shifts with changes in culture, immigration, and ideas about genetics. So who counts as white has changed over time. In the 1910s and 1920s, the last time immigrants were such a large share of the American population, there were furious arguments over how to categorize newcomers from Europe.

But eventually, the immigrants from eastern and southern Europe came to be considered white.

That is because race is about power, not biology, said Charles King, a political science professor at Georgetown University.

“The closer you get to social power, the closer you get to whiteness,” said Dr. King, author of a coming book on Franz Boas, the early 20th-century anthropologist who argued against theories of racial difference. The one group that was never allowed to cross the line into whiteness was African-Americans, he said — the long-term legacy of slavery.

To Richard Alba, a sociologist at the City University of New York, the Census Bureau’s projections seemed stuck in an outdated classification system. The bureau assigns a nonwhite label to most people who are reported as having both white and minority ancestry, he said. He likened this to the one-drop rule, a 19th-century system of racial classification in which having even one African ancestor meant you were black.

Asked for a response to Dr. Alba’s critique, a Census Bureau spokesman said in an email that “we constantly consult with stakeholders, and scholars, including Richard Alba and other federal agencies to improve our techniques, methodologies, and testing of population projections.”

William Frey, a demographer at the Brookings Institution, argued that the Census Bureau was doing the best that it could at a time when society was changing quickly. He was skeptical that today’s Asians and Hispanics were analogous to the white ethnic Americans of the 20th century, and believed that a less conservative count would not do much to change the bigger picture. Besides, it is not the job of academics to protect people from demographic change, he said.